The Global Integrated Monetary and Fiscal Model

(GIMF)

Technical Appendix

Michael Kumhof, International Monetary Fund

Douglas Laxton, International Monetary Fund

March 30, 2009

1

1 MODEL OVERVIEW

The world consists of Ñ countries. The domestic economy is indexed by 1 and foreign economies

by j = 2, ..., Ñ . In our exposition we will ignore country indices except when interactions between

multiple countries are concerned. It is understood that all parameters except population growth n and

technology growth g can differ across countries. Figure 1 illustrates the flow of goods and factors for

the two country case.

Countries are populated by two types of households, both of which consume final retailed output

and supply labor to unions. First, there are overlapping generations households with finite planning

horizons as in Blanchard (1985). Each of these agents faces a constant probability of death (1−θ(j))in each period, which implies an average planning horizon of 1/ (1− θ(j)).1 In each period,N(j)nt(1−ψ(j))

(1− θ(j)

n

)of such individuals are born, where N(j) indexes absolute population

sizes in period 0 and ψ(j) is the share of liquidity constrained agents. Second, there are liquidity

constrained households who do not have access to financial markets, and who consequently are forced

to consume their after tax income in every period. The number of such agents born in each period is

N(j)ntψ(j)(

1− θ(j)n)

. Aggregation over different cohorts of agents implies that the total numbers

of agents in country j is N(j)nt. For computational reasons we do not normalize world population

to one, especially when we analyze a small open economy. In that case we assume N(1) = 1, and

set N(j) such that N(1)/ΣÑj=2N(j) equals the share of country 1 agents in the world population. In

addition to the probability of death households also experience labor productivity that declines at a

constant rate over their lifetimes. This simplified treatment of lifecycle income profiles is justified

by the absence of explicit demographics in our model, and adds another powerful channel through

which fiscal policies can have non-Ricardian effects. Households of both types are subject to uniform

labor income, consumption and lump-sum taxes. We will denote variables pertaining to these two

groups of households by OLG and LIQ.

1 In general we allow for the possibility that agents may be more myopic than what would

be suggested by a planning horizon based on a biological probability of death.

2

Firms are managed in accordance with the preferences of their owners, myopic OLG households,

and they therefore also have finite planning horizons. Each country’s primary production is carried

out by manufacturers producing tradable and nontradable goods. Manufacturers buy capital services

from capital goods producers (in GIMF without Financial Accelerator) or from entrepreneurs (in

GIMF with Financial Accelerator), labor from monopolistically competitive unions, and oil from the

world oil market. They are subject to nominal rigidities in price setting as well as real rigidities in

labor hiring and in the use of oil. Capital goods producers are subject to investment adjustment costs.

Entrepreneurs finance their capital holdings using a combination of external and internal financing.

A capital income tax is levied on capital goods producers (in GIMF without Financial Accelerator)

or on entrepreneurs (in GIMF with Financial Accelerator). Unions are subject to nominal wage

rigidities and buy labor from households. Manufacturers’ domestic sales go to domestic distributors.

Their foreign sales go to import agents that are domestically owned but located in each export

destination country. Import agents in turn sell their output to foreign distributors. When the

pricing-to-market assumption is made these import agents are subject to nominal rigidities in foreign

currency. Distributors first assemble nontradable goods and domestic and foreign tradable goods,

where changes in the volume of imported inputs are subject to an adjustment cost. This private sector

output is then combined with a publicly provided capital stock (infrastructure) as an essential further

input. This capital stock is maintained through government investment expenditure that is financed

by tax revenue. The combined final domestic output is then sold to consumption goods producers,

investment goods producers, and import agents located abroad. Consumption and investment goods

producers in turn combine domestic and foreign output to produce final consumption and investment

goods. Foreign output is purchased through a second set of import agents that can price to the

domestic market, and again changes in the volume of imported goods are subject to an adjustment

cost. This second layer of trade at the level of final output is critical for allowing the model to produce

the high trade to GDP ratios typically observed in small, highly open economies. Consumption goods

output is sold to retailers and the government, while investment goods output is sold domestic capital

goods producers and the government. Consumption and investment goods producers are subject

to another layer of nominal rigidities in price setting. This cascading of nominal rigidities from

upstream to downstream sectors has important consequences for the behavior of aggregate inflation.

3

Retailers, who are also monopolistically competitive, face real instead of nominal rigidities. While

their output prices are flexible they find it costly to rapidly adjust their sales volume. This feature

contributes to generating inertial consumption dynamics.2

The world economy experiences a constant positive trend technology growth rate g = Tt/Tt−1,

where Tt is the level of labor augmenting world technology, and a constant positive population growth

rate n. When the model’s real variables, say xt, are rescaled, we divide by the level of technology

Tt and by population, but for the latter we divide by nt only, meaning real figures are not in per

capita terms but rather in absolute terms adjusted for technology and population growth. We use the

notation x̌t = xt/(Ttnt), with the steady state of x̌t denoted by x̄. An exception to this is quantities

of labor, which are only rescaled by nt.

Asset markets are incomplete. There is complete home bias in government debt, which takes

the form of nominally non-contingent one-period bonds denominated in domestic currency. The

only assets traded internationally are nominally non-contingent one-period bonds denominated in

the currency of Ñ . There is also complete home bias in ownership of domestic firms. In addition

equity is not traded in domestic financial markets, instead households receive lump-sum dividend

payments. This assumption is required to support our assumption that firm and not just household

preferences feature myopia.

2 The alternative of using habit persistence to generate consumption inertia is not availablein our setup. This is because we face two constraints in our choice of household preferences. The first is that

preferences must be consistent with balanced growth. The second is the necessity of beingable to aggregate across generations of households. We are left with preferences that, while commonly used,

do not allow for a powerful role of habit persistence.

4

OLG HH(HO) OLG HH(RW)LIQHH(HO) LIQHH(RW)

lOLG lLIQ

L=U

Unions (HO)

cOLG cLIQ

Inv. Producers (HO)

ZD

NTG(HO)

Manufacturers

YN ZT

YTF

YT

IN IT

KTKNUN UT

G

Gov’t (HO)

Import Adj. Costs

Sticky Prices Sticky Prices

Sticky

Wages

Unions (RW)

L*=U*

l*OLGc*OLGl*LIQc*LIQ

C*

Z*D

U*N

Tradables (RW)Manufacturers

Nontradables (RW)ManufacturersI

*NI*T

K*T K*N

Z*T Y*N

Y*TF

Y*TSticky Prices Sticky Prices

Retailers (HO)

Sticky Quantities

KG

YTH Y*TH

Gov’t (RW)

G*

K*GYA

YIH YIF

Sticky Prices ImportAgents(HO)

ImportAgents(RW) Y

*A

Y*IF Y*IH

Sticky Prices

ZN Z*N

ImportAgents

(HO)

Import

Agents(HO)YCH

Cons. Producers (HO)

YCF

CretSticky Prices

ImportAgents

(RW)

ImportAgents(RW)

Inv. Producers (RW)

Y*CF

Sticky Prices

Y*CH

Cons. Producers(RW)

Distributors

(HO)

Distributors

(RW)

Import Adj. Costs

GIMF

TG(HO)

Manufacturers

EPs EPs EPsEPs U*T

Oil (HO)

XN XT

WorldOil Market

X*T

Oil (RW)

Inv.Adj.Costs Inv.Adj.Costs

X*N

Sticky Wages

XC

C Retailers (RW)

XC* Cret*

Figure 1

5

2 Overlapping Generations Households

We first describe the optimization problem of OLG households. A representative member of this

group and of age a derives utility at time t from consumption cOLGa,t , leisure (SLt − �OLGa,t ) (where SLt

is the stochastic time endowment, which has a mean of one but which can itself be a function of the

business cycle, see below), and real balances (Ma,t/PRt ) (where P

Rt is the retail price index). The

lifetime expected utility of a representative household of age a at time t has the form

Et

∞∑

s=0

(βtθ)s

[1

1− γ((cOLGa+s,t+s

)ηOLG (SLt − �OLGa+s,t+s

)1−ηOLG)1−γ+

um

1− γ

(Ma+s,t+sPRt+s

)1−γ]

,

(1)

where Et is the expectations operator, θ < 1 is the degree of myopia, γ > 0 is the coefficient of

relative risk aversion, 0 < ηOLG < 13, um > 0, and βt is the (stochastic) discount factor. As

for money demand, in the following analysis we will only consider the case of the cashless limit

advocated by Woodford (2003), where um −→ 0. As a result the optimality conditions for moneywill be ignored throughout our analysis. Note that this does not involve a great loss of generality in

our case, and in fact it has one major advantage. The reason is that the combination of separable

money in the utility function and monetary policy specified as an interest rate rule implies that

the money demand equation becomes redundant and that inflation is not directly distortionary for

the consumption-leisure decision. But money also has a fiscal role through the government budget

constraint, and any reduction in inflation tax revenue must be accompanied by an offsetting increase

in other forms of distortionary taxation.4 Because of this indirect distortionary effect, an increase in

inflation in this model would actually reduce overall distortions unless we consider the case of the

cashless limit, in which case inflation causes no distortions in either direction.

3 For flexible model calibration we allow for the possibility that OLG households attacha different weight ηOLG to consumption than liquidity constrained households. This allows us to model bothgroups as working during an equal share of their time endowment in steady state, while OLGhouseholds have much higher consumption due to their accumulated wealth.4 Except for the special case of lump-sum taxation.

6

Consumption cOLGa,t is given by a CES aggregate over retailed consumption goods varieties

cOLGa,t (i), with elasticity of substitution σR:

cOLGa,t =

(∫ 1

0

(cOLGa,t (i)

)σR−1σR di

) σRσR−1

. (2)

This gives rise to a demand for individual varieties

cOLGa,t (i) =

(PRt (i)

PRt

)−σRcOLGa,t , (3)

where PRt (i) is the retail price of variety i, and the aggregate retail price level PRt is given by

PRt =

(∫ 1

0

(PRt (i)

)1−σRdi

) 11−σR

. (4)

A household can hold two types of bonds. The first bond type is domestic bonds denominated

in domestic currency. Such bonds are issued either by the domestic government Ba,t or, in the case

of GIMF with a Financial Accelerator, by banks lending to the nontradables or tradables sector,

BNa,t +BTa,t. The second bond type is foreign bonds denominated in the currency of country Ñ , Fa,t.

The nominal exchange rate vis-a-vis Ñ is denoted by Et, and EtFa,t are nominal net foreign assetholdings in terms of domestic currency. In each case the time subscript t denotes financial claims

held from period t to period t + 1. Gross nominal interest rates on domestic and foreign currency

denominated assets held from t to t + 1 are it/(1 + ξbt) and it(Ñ)(1 + ξ

ft ). For domestic bonds,

it is the nominal interest rate paid by the domestic government and ξbt is a domestic risk premium,

with ξbt < 0 characterizing a situation where the private sector faces a larger marginal funding rate

than the public sector. For foreign bonds, it(Ñ) is the nominal interest rate determined in Ñ , and ξft

is a foreign exchange risk premium. For country Ñ , it = it(Ñ) and ξft = 0. Both risk premia are

external to the household’s asset accumulation decision, and are payable to a financial intermediary

that redistributes the proceeds in a lump-sum fashion either to foreigners or to domestic households.

The functional form of the foreign exchange risk premium is given by

ξft = y1 +y2(

cagdpfiltt − y4)y3 + S

fxt , (5)

cagdpfiltt = Et

(Σkcahk=kcal

100cat+jgdpt+j

)/ (kcah − kcal + 1) , (6)

7

where Sfxt is a mean zero risk premium shock, y1 − y4 are parameters, y1 is constrained to generatea zero premium at a zero current account by the condition y1 = −y2/ (−y4)y3 , and cagdpfiltt is amoving average of past and future current account to GDP ratios, with kcah the maximum lead and

kcal the maximum lag. We have found this functional form to be more suitable for applied work than

conventional linear specifications because it is asymmetric, allowing for a steeply increasing risk

premium at large current account deficits.

The functional form of the domestic risk premium can similarly be made to depend on the

government debt to GDP ratio when it is intended to highlight the effect of government borrowing

levels on domestic interest rates. But it can also be treated as an exogenous stochastic process when

the emphasis is on shocks to the interest rate margin between the policy rate and the rate at which the

private sector can access the domestic capital market. For example, recent financial markets events

may be partly characterized by a persistent negative shock to ξbt .

Participation by households in financial markets requires that they enter into an insurance contract

with companies that pay a premium of(1−θ)θ on a household’s financial wealth for each period in

which that household is alive, and that encash the household’s entire financial wealth in the event of

his death.5

Apart from returns on financial assets, households also receive labor and dividend income.

Households sell their labor to “unions” that are competitive in their input market and monopolistically

competitive in their output market, vis-à-vis manufacturing firms. The productivity of a household’s

labor declines throughout his lifetime, with productivity Φa,t = Φa of age group a given by

Φa = κχa , (7)

where χ < 1. The overall population’s average productivity is assumed without loss of generality to

be equal to one. Household pre-tax nominal labor income is therefore WtΦa,t�OLGa,t . Dividends

are received in a lump-sum fashion from all firms in the nontradables (N) and tradables (T )

manufacturing sectors, from the distribution (D), consumption goods distribution (C) and investment

goods distribution (I) sectors, from the retail (R) sector and the import agent (M) sector, from

all unions (U ) in the labor market, from domestic (X) and foreign (F ) raw materials producers,

5 The turnover in the population is assumed to be large enough that the income receipts

of the insurance companies exactly equal their payouts.

8

from capital goods producers (K), and from entrepreneurs (EP ) (only in GIMF with Financial

Accelerator), with after-tax nominal dividends received from firm/union i denoted by Dja,t(i),

j = N,T,D,C, I, R, U,M,X,F,K,EP . OLG households are liable to pay lump-sum transfers

τOLGTa,t to the government, which in turn redistributes them to the relatively less well off LIQ agents.

Household labor income is taxed at the rate τL,t, and consumption is taxed at the rate τ c,t. In addition

there are lump-sum taxes τ lsa,t and transfers Υa,t paid to/from the government.6 It is assumed that

retailers face costs of rapidly adjusting their sales volume. To limit these costs they therefore offer

incentives (or disincentives) that are incorporated into the effective retail purchase price PRt . The

consumption tax τ c,t is however assumed to be payable on the pre-incentive price PCt .

7 PCt is

the marginal cost of retailers, who combine the output of consumption goods producers, with price

level Pt, with raw materials used directly by consumers, with price level PXt . We choose Pt as our

numeraire, and denote the real wage bywt = Wt/Pt, the relative price of any good x by pxt = P

xt /Pt,

gross inflation for any good x by πxt = Pxt /P

xt−1, and gross nominal exchange rate depreciation by

εt = Et/Et−1.8

The household’s budget constraint in nominal terms is

PRt cOLGa,t + P

Ct cOLGa,t τ c,t + Ptτ

lsa,t + Ba,t + B

Na,t + B

Ta,t + EtFa,t (8)

=1

θ

[it−1

(1 + ξbt−1)

(Ba−1,t−1 + B

Na−1,t−1 + B

Ta−1,t−1

)+ it−1(Ñ)EtFa−1,t−1(1 + ξft−1)

]

+WtΦa,t�OLGa,t (1− τL,t) +

∑

j=N,T,D,C,I,R,U,M,X,F,K,EP

1∫

0

Dja,t(i)di− τOLGTa,t + PtΥa,t .

The OLG household maximizes (1) subject to (2), (7) and (8). The derivation of the first-

order conditions for each generation, and aggregation across generations, is discussed in detail in

Appendices A and B. Aggregation takes account of the size of each age cohort at the time of birth, and

of the remaining size of each generation. Using the example of overlapping generations households’

6 It is sometimes convenient to keep these two items separate when trying to account for

a country’s overall fiscal structure, and when allowing for targeted transfers to LIQ agents.7 Without this assumption consumption tax revenue could become too volatile in the short run.8 We adopt the convention throughout the paper that all nominal price level variables are

written in upper case letters, and that all relative price variables are written in lower case letters.

9

consumption, we have

cOLGt = Nnt(1− ψ)

(1− θ

n

)Σ∞a=0

(θ

n

)acOLGa,t . (9)

This also has implications for the intercept parameter κ of the age-specific productivity distribution.

Under the assumption of an average productivity of one, and for given parameters χ and θ, we

obtain κ = (n − θχ)/(n − θ). Several of the optimality conditions that need to be aggregatedare, or are derived from, nonlinear Euler equations. In such conditions, aggregation requires

nonlinear transformations that are only valid under certainty equivalence. Tractable aggregate

consumption optimality conditions therefore only exist for the cases of perfect foresight and of first-

order approximations. For our purposes this is not problematic as all applications of GIMF will use

at most log-linear approximations. However, for the purpose of exposition we find it preferable to

present optimality conditions in nonlinear form. We therefore adopt the notation Ẽt to denote an

expectations operator that is understood in this fashion.

The first-order conditions for the goods varieties and for the consumption/leisure choice are given

by

čOLGt (i) =

(PRt (i)

PRt

)−σRčOLGt , (10)

čOLGtN(1− ψ)SLt − �̌OLGt

=ηOLG

1− ηOLG w̌t(1− τL,t)

(pRt + pCt τ c,t)

. (11)

The arbitrage condition for foreign currency bonds (the uncovered interest parity relation) is given

by

it = it(Ñ)(1 + ξft )(1 + ξ

bt)Ẽtεt+1 . (12)

The consumption Euler equation on the other hand cannot be directly aggregated across generations.

For each generation we have

Etca+1,t+1 = Etjtca,t , (13)

jt =

(β

řt+1

) 1γ

(pRt + p

Ct τ c,t

pRt+1 + pCt+1τ c,t+1

) 1γ

(

χgw̌t+1(1− τL,t+1)(pRt + pCt τ c,t)w̌t(1− τL,t)(pRt+1 + pCt+1τ c,t+1)

)(1−ηOLG)(1− 1γ)

.

(14)

10

Here we have used the notation

řt =it−1

πt(1 + ξbt−1

) =rt(

1 + ξbt−1) . (15)

We introduce some additional notation. The production based real exchange rate vis-a-vis Ñ is

et = (EtPt(Ñ))/Pt, where Pt(Ñ) is the price of final output in Ñ . We adopt the convention thateach nominal asset is deflated by the final output price index of the currency of its denomination, so

that real domestic bonds are bt = Bt/Pt and real foreign bonds are ft = Ft/Pt(Ñ). The real interest

rate in terms of final output payable by the government is rt = it/πt+1, while the real interest

rate payable by the private sector is řt = (it/πt+1) /(1 + ξbt

). The subjective and market nominal

discount factors are given by

R̃t,s = Πsl=1

θ(1 + ξbt+l−1

)

it+l−1for s > 0 ( = 1 for s = 0) , (16)

Rt,s = Πsl=1

(1 + ξbt+l−1

)

it+l−1for s > 0 ( = 1 for s = 0) , (17)

and the subjective and market real discount factors by

r̃t,s = Πsl=1

θ

řt+l−1for s > 0 ( = 1 for s = 0) , (18)

rt,s = Πsl=1

1

řt+l−1for s > 0 ( = 1 for s = 0) . (19)

In each case the subjective discount factor incorporates an agent’s probability of economic death,

which ceteris paribus makes him value near term receipts more highly than receipts in the distant

future.

We now discuss a key condition of GIMF, the optimal aggregate consumption rule of OLG

households. The derivation of this condition is algebraically complex and is therefore presented

in Appendix C. The final result expresses current aggregate consumption of OLG households as

a function of their real aggregate financial wealth fwt and human wealth hwt, with the marginal

propensity to consume of out of wealth given by 1/Θt. Human wealth is in turn composed of hwLt ,

the expected present discounted value of households’ time endowments evaluated at the after-tax

real wage, and hwKt , the expected present discounted value of capital or dividend income net of

lump-sum transfer payments to the government. After rescaling by technology we have

11

čOLGt Θt = f̌wt + ȟwt , (20)

where

f̌wt =1

πtgn

[it−1(

1 + ξbt−1) b̌t−1 + it−1(Ñ)(1 + ξ

ft−1)εtf̌t−1et−1

]

, (21)

ȟwt = ȟwLt + ȟw

Kt , (22)

ȟwLt =(N(1− ψ)(w̌t(1− τL,t)SLt )

)+ Ẽt

θχg

řt+1ȟwLt+1 , (23)

ȟwKt =(

Σj=N,T,D,C,I,R,U,M,X,F,K,EP ďjt − τ̌OLGT,t + Υ̌OLGt − τ̌ ls,OLGt

)+ Ẽt

θg

řt+1ȟwKt+1 , (24)

Θt =pRt + p

Ct τ c,t

ηOLG+ Ẽt

θjtřt+1

Θt+1 . (25)

The intuition of (20) is key to GIMF. Financial wealth (21) is equal to the domestic government’s and

foreign households’ current financial liabilities. For the government debt portion, the government

services these liabilities through different forms of taxation, and these future taxes are reflected in the

different components of human wealth (22) as well as in the marginal propensity to consume (25).

But unlike the government, which is infinitely lived, an individual household factors in that he might

not be around by the time higher future tax payments fall due. Hence a household discounts future

tax liabilities by a rate of at least řt/θ, which is higher than the market rate řt, as reflected in the

discount factors in (23), (24) and (25). The discount rate for the labor income component of human

wealth is even higher at řt/θχ, due to the decline of labor incomes over individuals’ lifetimes.

A fiscal consolidation through higher taxes represents a tilting of the tax payment profile from the

more distant future to the near future, so as to effect a reduction in the debt stock. The government

has to respect its intertemporal budget constraint in effecting this tilting, and this means that the

expected present discounted value of its future primary surpluses has to remain equal to the current

debt it−1bt−1/πt when future surpluses are discounted at the market interest rate rt. But when

individual households discount future taxes at a higher rate than the government, the same tilting

12

of the tax profile represents a decrease in human wealth because it increases the expected value of

future taxes for which the household expects to be responsible. This is true both for the direct effect

of labor income taxes on labor income receipts, and for the indirect effect of corporate taxes on

dividend receipts. For a given marginal propensity to consume, these reductions in human wealth

lead to a reduction in consumption. Note that with ξbt < 0 this effect is not only due to myopia but

also to the borrowing spread between the public and private sectors.

The marginal propensity to consume 1/Θt is, in the simplest case of logarithmic utility and

exogenous labor supply, equal to (1 − βθ). For the case of endogenous labor supply, householdwealth can be used to either enjoy leisure or to generate purchasing power to buy goods. The main

determinant of the split between consumption and leisure is the consumption share parameter ηOLG,

which explains its presence in the marginal propensity to consume (25). While other forms of taxation

affect the different components of wealth, the time profile of consumption taxes affects the marginal

propensity to consume, reducing it with a balanced-budget shift of such taxes from the future to the

present. The intertemporal elasticity of substitution 1/γ is another key parameter for the marginal

propensity to consume. For the conventional assumption of γ > 1 the income effect of an increase

in the real interest rate r is stronger than the substitution effect and tends to increase the marginal

propensity to consume, thereby partly offsetting the contractionary effects of a higher r on human

wealth ȟwt. Larger γ therefore tends to give rise to larger interest rate changes in response to fiscal

shocks.

Finally, we allow for a variable utility value of a household’s time endowment. What we have

in mind is that as households increase their consumption relative to the balanced growth path, this

increases their sense of well-being not only directly, but also indirectly by making them value their

time endowment more highly. Consequently they are willing to work more and still derive an

increased utility value from the remaining leisure. This effect is however temporary in that the rate

of increase in the value of their time endowment increases in the rate of increase in (normalized)

consumption relative to a moving average of past (normalized) consumption, so that a permanent

increase in consumption does not ultimately lead to a permanent increase in the time endowment.

13

Furthermore, the strength of this effect can be calibrated by choosing the parameter toil. We have

SLt − 1 = toil ∗(čOLGt

čfiltt− 1)

, (26)

čfiltt =(

Σkc

j=1ωcj čOLGt−j

)/kc , (27)

where Σkc

j=1ωcj = 1. We introduce this effect because it addresses a common problem with household

preferences in calibrated DSGE models. Namely, as can be seen by inspecting (11), the model would

ordinarily suggest that an increase in consumption demand is accompanied by an increase in the

demand for leisure, and therefore by lower labor supply, a lower marginal productivity of capital,

and therefore lower investment. To interpret the fact that in the data consumption, investment and

employment show a positive comovement, the model would have to assign a major role to supply

shocks that do generate such comovement. This is unsatisfactory because intuition suggests that

standard demand shocks also generate a positive comovement. The specification (26), depending on

the parameter toil, generates such comovement for all shocks by generating a positive comovement

between consumption and labor supply.

3 Liquidity Constrained Households

The objective function of liquidity constrained (LIQ) households is assumed to be nearly

identical to that of OLG households:9

Et

∞∑

s=0

(βθ)s[

1

1− γ

((cLIQa+s,t+s

)ηLIQ (SLt − �LIQa+s,t+s

)1−ηLIQ)1−γ]

, (28)

cLIQa,t =

(∫ 1

0

(cLIQa,t (i)

)σR−1σR di

) σRσR−1

. (29)

These agents can consume at most their current income, which consists of their after tax wage income

plus government transfers τLIQTa,t . Their budget constraint is

PRt cLIQa,t + P

Ct cLIQa,t τ c,t �WtΦa,t�

LIQa,t (1− τL,t) + τ

LIQTa,t

+ ΥLIQa,t − τls,LIQa,t . (30)

9 The distinction of generations could be dropped as all agents must act identically.

14

The aggregated first-order conditions for this problem, after rescaling by technology, are

čLIQt (i) =

(PRt (i)

PRt

)−σRčLIQt , (31)

čLIQt (pRt + p

Ct τ c,t) = w̌t�

LIQt (1− τL,t) + τ̌

LIQT,t + Υ̌

LIQt − τ̌

ls,LIQt , (32)

čLIQt

NψSLt − �̌LIQt=

ηLIQ

1− ηLIQ w̌t(1− τL,t)

(pRt + pCt τ c,t)

. (33)

GIMF also allows for an alternative version where equation (33) is dropped and is replaced with an

exogenous labor supply, the so-called “rule of thumb consumer”.

4 Aggregate Household Sector

To obtain aggregate consumption demand and labor supply we simply add the respective

optimality quantities of the different consumers in the economy. For GIMF without a Financial

Accelerator these are OLG and LIQ households:

Čt = čOLGt + č

LIQt , (34)

Ľt = �̌OLGt + �̌

LIQt . (35)

15

5 Manufacturers

There is a continuum of manufacturing firms indexed by i ∈ [0, 1] in two separate manufacturingsectors indexed by J ∈ {N,T}, whereN represents nontradables and T tradables. For prices in thesetwo sectors we introduce a slightly different index J̃ ∈ {N,TH}, because the index T for pricesis reserved for a different goods aggregate produced by distributors (see below). Manufacturers

buy labor inputs from unions and capital inputs from capital goods producers. Sector N and T

manufacturers sell to domestic distributors, and sector T manufacturers also sell to import agents in

foreign countries, who in turn sell to distributors in those countries.10 Manufacturers are perfectly

competitive in their input markets and monopolistically competitive in the market for their output.

Their price setting is subject to nominal rigidities. We first analyze the demands for their output, then

turn to their technology, and finally describe their optimization problem.

Demands for manufacturers’ output varieties are given by

Y Jt (z) =

1∫

0

Y Jt (z, i)σJ−1

σJ di

σJσJ−1

, Y TXt (1, j, z) =

1∫

0

Y TXt (1, j, z, i)σJ−1

σJ di

σJσJ−1

,

(36)

where Y Jt (z, i) and YJt (z) are variety i and total demands from domestic distributor z in sector J ,

and Y TXt (1, j, z, i) and YTXt (1, j, z) are variety i and total demands for exports from country 1 to

import agent z in country j. Cost minimization by distributors and import agents generates demands

for varieties

Y Jt (z, i) =

(P J̃t (i)

P J̃t

)−σJY Jt (z) , Y

TXt (1, j, z, i) =

(PTHt (i)

PTHt

)−σJY TXt (1, j, z) , (37)

with price indices defined as

P J̃t =

1∫

0

P J̃t (i)1−σJdi

1

1−σJ

. (38)

10 There are also some small sales of aggregate manufacturing output back to manufacturing

firms, related to manufacturers’ need for resources to pay for adjustment costs.

16

The aggregate demand for variety i produced by sector J can be derived by simply integrating over

all distributors, import agents and all other sources of manufacturing output demand. We obtain

ZJt (i) =

(P J̃t (i)

P J̃t

)−σJZJt , (39)

where ZJt (i) and ZJt remain to be specified by way of market clearing conditions for manufacturing

goods.

The technology of each manufacturing firm differs depending on whether the raw materials sector

is included. If it is included, the technology is given by a CES production function in capital KJt−1(i),

union labor UJt (i) and raw materials XJt (i), with elasticities of substitution ξZJ between capital and

labor, and ξXJ between raw materials and capital/labor. An adjustment cost GJX,t(i) makes fast

changes in raw materials inputs costly. Labor augmenting productivity is TtAJt , where A

Jt is a

country specific technology shock:11,12

ZJt (i) = F (KJt−1(i), U

Jt (i), X

Jt (i)) (40)

= T

((1− αXJt

) 1ξXJ

(MJt (i)

) ξXJ−1ξXJ +

(αXJt) 1ξXJ

(XJt (i)

(1−GJX,t(i)

)) ξXJ−1ξXJ

) ξXJξXJ−1

,

MJt (i) =

((1− αUJ

) 1ξZJ

(KJt−1(i)

) ξZJ−1ξZJ +

(αUJ) 1ξZJ

(TtA

Jt UJt (i)

) ξZJ−1ξZJ

) ξZJξZJ−1

.

If the raw materials sector is not included, the technology is given by a CES production function in

capital KJt (i) and union labor UJt (i), with elasticity of substitution ξZJ between capital and labor:

ZJt (i) = F (KJt−1(i), U

Jt (i)) (41)

= T

((1− αUJ

) 1ξZJ

(KJt−1(i)

) ξZJ−1ξZJ +

(αUJ) 1ξZJ

(TtA

Jt UJt (i)

) ξZJ−1ξZJ

) ξZJξZJ−1

.

We will from now on mostly ignore the version without raw materials sector, for which the optimality

conditions can be derived in the same fashion as below.

11 Note that, for the sake of clarity, we make a notational distinction between two types of

elasticities of substitution. Elasticities between continua of goods varieties, which give riseto market and pricing power, are denoted by a σ subscripted by the respective sectorial indicator.Elasticities between factors of production, both in manufacturing and in final goods distribution, are denotedby a ξ subscripted by the respective sectorial indicator.12 The factor T is a constant that can be set different from one to obtain different levelsof GDP per capita across countries.

17

Manufacturing firms are subject to three types of adjustment costs. First, quadratic inflation

adjustment costs GJP,t(i) are real resource costs that represent a demand for the output of sector J .

Following Ireland (2001) and Laxton and Pesenti (2003), they are quadratic in changes in the rate of

inflation rather than in price levels, which is essential in order to generate realistic inflation dynamics.

Compared to versions of the Calvo (1983) price setting assumption such adjustment costs have the

advantage of greater analytical tractability. We have:

GJP,t(i) =φP J

2ZJt

P J̃t (i)

P J̃t−1(i)

P J̃t−1

P J̃t−2

− 1

2

. (42)

To allow a flexible choice of inflation adjustment costs we also allow for a version of Rotemberg

(1982) sticky prices, whereby deviations of the actual inflation rate from the inflation target π̄t are

costly. These may sometimes be preferable when working with a fixed exchange rates model, where

sticky inflation can give rise to strong cycles. These costs are given by13

GJP,t(i) =φP J

2ZJt

(P J̃t (i)

P J̃t−1(i)− π̄t

)2. (43)

Second, adjustment costs on raw materials inputs are given by14

GJX,t(i) =φJX2

((XJt (i)/ (gn))−XJt−1

XJt−1

)2, (44)

the term gn enters to ensure that adjustment costs are zero along the balanced growth path.

Third, adjustment costs on labor hiring are given by

GJU,t(i) =φU2UJt

((UJt (i)/n)− UJt−1(i)

UJt−1(i)

)2. (45)

These costs are somewhat less common in the business cycle literature, and are only included as an

option that can be switched off by setting φU = 0.

It is assumed that each firm pays out each period’s after tax nominal net cash flow as dividends

DJt (i). It maximizes the expected present discounted value of dividends. The discount rate it applies

in this maximization includes the parameter θ so as to equate the discount factor of firms θ/řt with

the pricing kernel for nonfinancial income streams of their owners, myopic households, which equals

13 In all other instances of nominal rigidities that follow, GIMF offers this as one option.It will however not be mentioned again in this document.14 Note that, unlike other adjustment costs, this expression treats lagged inputs as external.

This has proved more useful than the alternatives in our applied work.

18

βθEt (λa+1,t+1/λa,t). This equality follows directly from OLG households’ first order condition for

government debt holdings λa,t = βEt

(λa+1,t+1

itπt+1(1+ξbt)

).

Pre-tax net cash flow equals nominal revenue P J̃t (i)ZJt (i) minus nominal cash outflows. The

latter include the wage bill VtUJt (i), where Vt is the aggregate wage rate charged by unions,

spending on raw materials PXt XJt (i), where P

Xt is the price of raw materials, and the cost of

capital RJk,tKJt−1(i), where R

KJ

t is the nominal rental cost of capital in sector J , with the real

cost denoted rJk,t. Other components of pre-tax cash flow are price adjustment costs PJ̃t GJP,t(i)

that represent a demand for sectorial manufacturing output ZJt , labor adjustment costs VtGU,t(i)

that represent a demand for labor Lt, and a fixed cost PJ̃t Ttω

J . The fixed resource cost arises as

long as the firm chooses to produce positive output. Net output in sector J is therefore equal to

max(0, ZJt (i) − TtωJ). The fixed cost is calibrated to make the steady state shares of economicprofits, labor and capital in GDP consistent with the data. This becomes necessary because the

model counterpart of the aggregate income share of capital equals not only the return to capital

but also the profits of monopolistically competitive firms. With several layers of such firms the

profits share becomes significant, and the capital share parameter in the production function has to

be reduced accordingly, unless fixed costs are assumed. More importantly, the introduction of an

additional parameter determining fixed costs allows us to simultaneously calibrate not only capital

income shares and depreciation rates but also the investment to GDP ratio. This would otherwise be

impossible. We calibrate fixed costs by first noting that, in normalized form, steady state monopoly

profits equal ŽJt /σJ . We denote by sπ the share of these profits that remain after fixed costs have been

paid, and we will calibrate this parameter to obtain the desired investment to GDP ratio. We assume

that sπ is identical across the industries where fixed costs arise. Then fixed costs in manufacturing

are given by

ωJ =Z̄J

σJ(1− sπ) . (46)

The total after tax net cash flow or dividend of the firm is

DJt (i) = PJ̃t (i)Z

Jt (i)− VtUJt (i)−PXt XJt (i)−RJk,tKJt−1(i)−P J̃t TtωJ −P J̃t GJP,t(i)− VtGJU,t(i) .

(47)

19

The optimization problem of each manufacturing firm is

Max{P J̃t+s(i),UJt+s(i),KJt+s−1(i),XJt+s(i)}∞s=0

EtΣ∞s=0R̃t,sD

Jt+s(i) , (48)

subject to the definition of dividends (47), demands (39), production functions (40), and adjustment

costs (42)-(45). The first-order conditions for this problem are derived in some detail in Appendix D.

A key step is to recognize that all firms behave identically in equilibrium, so that P J̃t (i) = PJ̃t and

ZJt (i) = ZJt . Let λ

Jt denote the real marginal cost of producing an additional unit of manufacturing

output. Also, rescale the optimality conditions by technology and population as discussed above.

Then the condition for P J̃t (i) under sticky inflation is[σJ

σJ − 1λJt

pJ̃t− 1]

=φP J

σJ − 1

(πJ̃t

πJ̃t−1

)(πJ̃t

πJ̃t−1− 1)

(49)

−Etθgn

řt+1

φPJ

σJ − 1pJ̃t+1

pJ̃t

ŽJt+1ŽJt

(πJ̃t+1

πJ̃t

)(πJ̃t+1

πJ̃t− 1)

,

while under sticky prices we have[

σJσJ − 1

λJt

pJ̃t− 1]

=φPJ

σJ − 1πJ̃t

(πJ̃t − π̄t

)(50)

−Etθgn

řt+1

φPJ

σJ − 1pJ̃t+1

pJ̃t

ŽJt+1ŽJt

πJ̃t+1

(πJ̃t+1 − π̄t

).

The first order condition for labor demand UJt (i) is(λJtv̌tF̌ JU,t − 1

)= φU

(Ǔt

Ǔt−1

)(Ǔt − Ǔt−1Ǔt−1

)− θgnřt+1

φUv̌t+1v̌t

(Ǔt+1

Ǔt

)2(Ǔt+1 − Ǔt

Ǔt

), (51)

where F̌ JU,t is the marginal product of labor

F̌ JU,t = T((

1− αXJt)ŽJt

T M̌Jt

) 1ξXJ

AJt

(αUJ M̌

Jt

AJt ǓJt

) 1ξZJ

. (52)

The first order condition for raw materials demand XJt (i) is

pXt = λJt F̌JX,t , (53)

where F̌ JX,t is the marginal product of raw materials

F̌ JX,t = T

αXJtŽJt

T X̌Jt(

1−GJX,t)

1

ξXJ(

1−GJX,t − φJXX̌JtX̌Jt−1

(X̌Jt − X̌Jt−1

X̌Jt−1

))

. (54)

20

The first-order condition for capital demand is

rJk,t = λJt F̌JK,t , (55)

where F̌ JK,t is the marginal product of capital

F̌ JK,t = T((

1− αXJt)ŽJt

T M̌Jt

) 1ξXJ

((1− αUJ

)M̌Jt(

ǨJt−1/ (gn))

) 1ξZJ

. (56)

For the sake of completeness we add here the marginal products of labor and capital for the version

of GIMF without raw materials. They are

F̌ JU,t = T AJt(αUJ Ž

Jt

AJt ǓJt

) 1ξZJ

, (57)

F̌ JK,t = T( (

1− αUJ)ŽJt(

ǨJt−1/ (gn))

) 1ξZJ

. (58)

Rescaled aggregate dividends of firms in each sector are

ďJt =[pJ̃t Ž

Jt − v̌tǓJt − pXt X̌Jt − rJk,t

(ǨJt−1/ (gn)

)− v̌tǦJU,t − pJ̃t ǦJP,t − pJ̃t ωJ

]. (59)

We define aggregate capital and investment as

Ǐt = ǏNt + Ǐ

Tt , (60)

Ǩt = ǨNt + Ǩ

Tt . (61)

Finally, we turn to the market clearing conditions for nontradables and tradables. They equate

the output of each sector to the demands of distributors, of manufacturers themselves for fixed and

adjustment costs, and in the case of tradables to the demands of foreign import agents. We have15

ŽNt = Y̌Nt + ω

N + ǦNP,t + rcuNt + rbr

Nt + Š

N,nwyshkt , (62)

ŽTt (1) = Y̌THt (1) + ω

T (1) + ǦTP,t(1) + rcuTt + rbr

Tt + Š

T,nwyshkt + p̃

expt Σ

Ñj=2Y̌

TXt (1, j) , (63)

15 The tradables market clearing condition is reported for the example of country 1.

21

where rcuJt is the resource cost associated with variable capital utilization, rbrJt is the resource

cost due to bankruptcies (in GIMF with Financial Accelerator), and ŠJ,nwyshkt is the net effect of

entrepreneurs’ output destroying net worth shocks (in GIMF with Financial Accelerator). The term

p̃expt in the second market clearing condition refers to unit root shocks to the relative price of exported

goods. Specifically, tradables output is converted to exports Y̌ TXt using a technology that multiplies

tradables output by T expt = 1/p̃expt , where p̃

expt is a unit root shock with zero trend growth.

6 Capital Goods Producers

6.1 GIMF with Financial Accelerator

These agents produce the capital stock used by entrepreneurs in the nontradables and tradables

sectors, indexed as before by J ∈ {N,T}. They are competitive price takers. Capital goodsproducers are owned by households, who receive their dividends as lump-sum transfers. They

purchase previously installed capital K̃Jt−1 from entrepreneurs and investment goods IJt from

investment goods producers to produce new installed capital K̃Jt according to

K̃Jt = K̃Jt−1 + S

invt I

Jt , (64)

where Sinvt is an investment demand shock. They are subject to investment adjustment costs

GJI,t =φI2IJt

((IJt /(gn))− IJt−1

IJt−1

)2. (65)

The nominal price level of previously installed capital is denoted by QJt . Since the marginal rate of

transformation from previously installed to newly installed capital is one, the price of new capital

is also QJt . The optimization problem is to maximize the present discounted value of dividends by

choosing the level of new investment IJt :16

Max{IJt+s}∞s=0

EtΣ∞s=0R̃t,sD

KJ

t+s , (66)

DKJ

t = QJt

(K̃Jt−1 + S

invt I

Jt

)−QJt K̃Jt−1 − P It

(IJt + G

JI,t

). (67)

16 Any value of capital if profit maximizing.

22

The solution to this problem is

qJt Sinvt = p

It + φIp

It

(ǏJtǏJt−1

)(ǏJt − ǏJt−1ǏJt−1

)

−Etθgn

řt+1φIp

It+1

(ǏJt+1ǏJt

)2(ǏJt+1 − ǏJt

ǏJt

)

. (68)

The stock of physical capital evolves as

K̄Jt =(1− δJKt

)K̄Jt−1 + S

invt I

Jt . (69)

We allow for shocks to the deprecation rate of capital, which in the context of the Financial

Accelerator we will refer to as capital destroying net worth shocks:

δJKt = δ̄JK + S

nwkshkt . (70)

Physical capital K̄Jt is different from the capital rented by manufacturers KJt because the stock of

physical capital is subject to variable capital utilization uJt . The normalized relationship between

physical capital K̄J and capital used in manufacturing KJ is therefore given by

ǨJt−1 = uJt ǨJ

t−1 . (71)

The real value of dividends is given by

ďKJ

t = qJt Sinvt Ǐ

Jt − pIt

(ǏJt + Ǧ

JI,t

). (72)

We let ďKt = ďKN

t + ďKT

t , and also Ǐt = ǏNt + Ǐ

Tt , K̄t = K̄

Nt + K̄

Tt .

6.2 GIMF without Financial Accelerator

Capital goods producers produce the physical capital stock K̄Jt . They rent out lagged capital

K̄Jt−1 to manufacturers in the nontradables and tradables sectors J ∈ {N,T}, after deciding on therate of capital utilization uJt . They are competitive price takers and are subject to a capital income

tax. Capital goods producers are owned by households, who receive their dividends as lump-sum

transfers. The accumulation of the physical capital stock is given by

K̄Jt =(1− δJKt

)K̄Jt−1 + S

invt I

Jt . (73)

As before, we allow for shocks to the deprecation rate of capital as in (70). Investment goods IJt are

purchased from investment goods producers, where Sinvt is an investment demand shock. Investment

is subject to investment adjustment costs

GJI,t =φI2IJt

((IJt /(gn))− IJt−1

IJt−1

)2. (74)

23

After observing the time t aggregate shocks the entrepreneur decides on the time t level of capital

utilization uJt , and then rents out capital services KJt−1(j) = u

Jt K̄Jt−1(j). High capital utilization

gives rise to high costs in terms of sector J goods, according to the convex function a(uJt )K̄Jt−1(j),

where we specify the adjustment cost function as17

a(uJt ) =1

2φJaσ

Ja

(uJt)2

+ φJa(1− σJa

)uJt + φ

Ja

(σJa2− 1)

. (75)

The optimization problem is to maximize the present discounted value of dividends by choosing

the level of new investment IJt , the level of the physical capital stock K̄Jt , and the rate of capital

utilization uJt :

Max{IJt+s,K̄Jt+s,uJt+s}∞s=0

EtΣ∞s=0R̃t,sD

KJ

t+s , (76)

DKJ

t =((1− τk,t)

(RJk,tu

Jt − Pta(uJt )

)+ τk,tδ

JKtQ

Jt

)K̄Jt−1 − P It

(IJt + G

JI,t

)(77)

+QJt((

1− δJKt)K̄Jt−1 + S

invt I

Jt − K̄Jt

).

The first order conditions for investment demand IJt (i) and capital K̄Jt (i) are

qJt Sinvt = p

It + φIp

It

(ǏJtǏJt−1

)(ǏJt − ǏJt−1ǏJt−1

)

−Etθgn

řt+1φIp

It+1

(ǏJt+1ǏJt

)2(ǏJt+1 − ǏJt

ǏJt

)

, (78)

qJt =θ

řt+1Et[qJt+1(1− δJK) + (1− τk,t+1)

(uJt+1r

Jk,t+1 − a(uJt+1)

)+ τk,t+1δ

JKqJt+1

]. (79)

The first order condition for capital utilization is

rJk,t = φJaσJauJt + φ

Ja

(1− σJa

), (80)

and the resource cost associated with variable capital utilization is

rcuJt = a(uJt )

(ǨJ

t−1/ (gn)

)/pJ̃t . (81)

The real value of dividends is given by

ďKJ

t =((1− τk,t)

(rJk,tu

Jt − a(uJt )

)+ τk,tδ

JKtqJt

)K̄Jt−1 − pIt

(IJt + G

JI,t

). (82)

We let ďKt = ďKN

t + ďKT

t , and also Ǐt = ǏNt + Ǐ

Tt , K̄t = K̄

Nt + K̄

Tt .

17 This follows Christiano, Motto and Rostagno (2007), “Financial Factors in Business Cycles”.Papers where the model is linearized prior to solving it only require the elasticity σa of thefunction a(ut). Because GIMF is solved in nonlinear form we require a full functional form.

24

7 Entrepreneurs and Banks

Entrepreneurs in sectors J ∈ {N,T} purchase a capital stock from capital goods producers andrent it to manufacturers. Each entrepreneur j finances his time t capital holdings (at current market

prices) QJt K̄Jt (j) with a combination of his net worth N

Jt (j) and a bank loan B

Jt (j). His balance

sheet constraint is therefore given by

QJt K̄Jt (j) = N

Jt (j) + B

Jt (j) , (83)

or in real normalized terms byqJt Ǩ

J

t (j) = ňJt (j) + b̌

Jt (j) . (84)

After the capital purchase each entrepreneur draws an idiosyncratic shock which changes K̄Jt (j) to

ωJt+1K̄Jt (j) at the beginning of period t + 1, where ω

Jt+1 is a unit mean lognormal random variable

distributed independently over time and across entrepreneurs. The standard deviation of ln(ωJt+1),

σJt+1, is itself a stochastic process. While the realization of ωJt+1 is not known at the time the

entrepreneur makes his capital decision, the value of σJt+1 is known. The cumulative distribution

function of ωJt+1 is given by Pr(ωJt+1 ≤ x) = F Jt+1(x).

After observing the time t aggregate shocks the entrepreneur decides on the time t level of capital

utilization uJt , and then rents out capital services KJt−1(j) = u

Jt K̄Jt−1(j). High capital utilization

gives rise to high costs in terms of sector J goods, according to the convex function a(uJt )ωJt K̄Jt−1(j),

where we specify the adjustment cost function as

a(uJt ) =1

2φJaσ

Ja

(uJt)2

+ φJa(1− σJa

)uJt + φ

Ja

(σJa2− 1)

. (85)

The entrepreneur chooses uJt to solve

MaxuJt

[uJt r

Jk,t − a(uJt )

](1− τk,t)ωJt K̄Jt−1(j) , (86)

which has the solutionrJk,t = φ

JaσJauJt + φ

Ja

(1− σJa

). (87)

The resource cost associated with variable capital utilization is given by

rcuJt = a(uJt )

(ǨJ

t−1/ (gn)

)/pJ̃t , (88)

The entrepreneur’s real ex-post, after tax return to utilized capital is given by

retJk,t =

(uJt r

Jk,t − a(uJt ) +

(1− δJKt

)qJt

)− τk,t

(uJt r

Jk,t − a(uJt )− δJKtqJt

)

qJt−1. (89)

25

We assume that the entrepreneur receives a standard debt contract from the bank. This specifies a

loan amount BJt and a gross rate of interest iJB,t+1 to be paid if ω

Jt+1 is high enough. Entrepreneurs

who draw ωJt+1 below a cutoff level ω̄Jt+1 cannot pay this interest rate and go bankrupt. They must

hand over everything they have to the bank, but the bank can only recover a time-varying fraction

(1− µJt+1) of the value of such firms. The cutoff ω̄Jt+1 is defined as follows:

ω̄Jt+1retJk,t+1Q

Jt K̄Jt (j) = i

JB,t+1B

Jt (j) , (90)

where retJk,t+1 is the nominal ex-post after tax return to utilized capital. The bank finances its loans

to entrepreneurs by borrowing from households. We assume that the bank pays households a nominal

rate of return ǐt = it/(1 + ξbt

)that is not contingent on the realization of time t + 1 shocks. The

parameters of the entrepreneur’s debt contract are chosen to maximize entrepreneurial utility, subject

to zero profits in each state of nature for the bank and to the requirement that ǐt be non-contingent

on time t + 1 shocks. This implies that iJB,t+1 and ω̄Jt+1 are both functions of time t + 1 aggregate

shocks.

The bank’s zero profit or participation constraint is given by:18

(1− F (ω̄Jt+1)

)iJB,t+1B

Jt (j) +

(1− µJt+1

) ∫ ω̄Jt+1

0QJt K̄

Jt (j)ret

Jk,t+1ωf(ω)dω = ǐtB

Jt (j) . (91)

This states that the stochastic payoff to lending on the l.h.s. must equal the non-stochastic payment

to depositors on the r.h.s. in each state of nature. The first term on the l.h.s. is the nominal interest

income on loans for borrowers whose idiosyncratic shock exceeds the cutoff level, ωJt+1 ≥ ω̄Jt+1.The second term is the amount collected by the bank in case of the borrower’s bankruptcy, where

ωJt+1 < ω̄Jt+1. This cash flow is based on the return ret

Jk,t+1ω on capital investment Q

Jt K̄Jt (j), but

multiplied by the factor(1− µJt+1

)to reflect a proportional bankruptcy cost µJt+1. Next we rewrite

(91) by using (90) and (83):

[(1− F (ω̄Jt+1)

)ω̄Jt+1 +

(1− µJt+1

) ∫ ω̄Jt+1

0ωf(ω)dω

]

retJk,t+1QJt K̄Jt (j) (92)

= ǐtQJt K̄Jt (j)− ǐtNJt (j) .

18 Note the absence of expectations operators because this equation has to hold in each state

of nature. Likewise for subsequent equations.

26

We adopt a number of definitions that simplify the following derivations. First, note that capital

earnings are given by retJk,t+1QJt K̄Jt (j). The lender’s gross share in capital earnings is then defined

as

Γ(ω̄Jt+1) ≡∫ ω̄Jt+1

0ωJt+1f(ω

Jt+1)dω

Jt+1 + ω̄

Jt+1

∫ ∞

ω̄Jt+1

f(ωJt+1)dωJt+1 , (93)

while his monitoring costs share in capital earnings is given by µJt+1G(ω̄Jt+1), where

G(ω̄Jt+1) =

∫ ω̄Jt+1

0ωJt+1f(ω

Jt+1)dω

Jt+1 . (94)

The lender’s net share in capital earnings is therefore Γ(ω̄Jt+1) − µJt+1G(ω̄Jt+1). The entrepreneur’sshare in capital earnings on the other hand is given by

1− Γ(ω̄Jt+1) =∫ ∞

ω̄Jt+1

(ωJt+1 − ω̄Jt+1

)f(ωJt+1)dω

Jt+1 . (95)

Using this notation and denoting the multiplier of the participation constraint by λt, the

entrepreneur’s optimization problem can be written as

MaxK̄Jt (j),ω̄

Jt+1

(1− Γ(ω̄Jt+1)

)retJk,t+1Q

Jt K̄Jt (j) (96)

+λt{(

Γ(ω̄Jt+1)− µJt+1G(ω̄Jt+1))retJk,t+1Q

Jt K̄Jt (j)− ǐtQJt K̄Jt (j) + ǐtNJt (j)

}.

Before deriving the optimality conditions we rewrite this expression by dividing through by ǐtNJt (j),

rewriting the resulting expression in terms of normalized variables, and finally replacing nominal

returns by real returns:

MaxǨ

J

t (j),ω̄Jt+1

(1− Γ(ω̄Jt+1)

) rětJk,t+1řt+1

qJt ǨJ

t (j)

ňJt (j)(97)

+λt

(Γ(ω̄Jt+1)− µJt+1G(ω̄Jt+1)

) rětJk,t+1řt+1

qJt ǨJ

t (j)

ňJt (j)− q

Jt ǨJ

t (j)

ňJt (j)+ 1

.

We let ΓJt+1 = Γ(ω̄Jt+1), G

Jt+1 = G(ω̄

Jt+1), Γ

′J,t+1 = ∂Γ

Jt+1/∂ω̄

Jt+1 and G

′J,t+1 = ∂G

Jt+1/∂ω̄

Jt+1.

We obtain the following first-order condition with respect to ω̄Jt+1:

−Γ′J,t+1rětJk,t+1řt+1

qJt ǨJ

t (j)

ňJt (j)+ λt

(Γ′J,t+1 − µJt+1G′J,t+1

) rětJk,t+1řt+1

qJt ǨJ

t (j)

ňJt (j)

= 0 , (98)

which implies

27

λt =Γ′J,t+1

Γ′J,t+1 − µJt+1G′J,t+1. (99)

The condition for the optimal loan contract, that is the first-order condition with respect to ǨJ

t (j),

can be written using (99) as19

(1− ΓJt

) rětJk,třt

+Γ′J,t

Γ′J,t − µJt G′J,t

{rětJk,třt

(ΓJt − µJt GJt

)− 1}

= 0 , (100)

where we have replaced time t+1 subscripts with time t subscripts everywhere because this condition

has to hold for each state of nature, that is it has to hold exactly ex-post. The normalized lender’s

zero profit condition is

qJt ǨJ

t

ňJt

rětJk,t+1řt+1

(ΓJt+1 − µJt+1GJt+1)

)− q

Jt ǨJ

t

ňJt+ 1 = 0 . (101)

Notice that we have omitted entrepreneur specific indices j for capital and net worth and replaced

them with the corresponding aggregate variables. This is because each entrepreneur faces the same

returns rětJk,t+1 and řt+1, and the same risk environment characterizing the functions Γ and G.

Aggregation of the model over entrepreneurs is then trivial because both borrowing and capital

purchases are proportional to the entrepreneur’s level of net worth.

A key problem for coding the Financial Accelerator version of GIMF in a standard software such

as TROLL and DYNARE consists of finding a closed form representation for the terms ΓJt , GJt and

their derivatives. In TROLL we can use the hard-wired (like e.g. LOG) PNORM function, which

is the c.d.f. of the standard normal distribution. In Appendix E we therefore derive the relevant

expressions in terms of PNORM, for which we use the notation Φ(.). We obtain the following set of

equations, starting with an auxiliary variable z̄Jt :

z̄Jt =ln(ω̄Jt ) +

12

(σJt)2

σJt, (102)

f(ω̄Jt)

=1√

2πω̄Jt σJt

exp

{−1

2

(z̄Jt)2}

, (103)

ΓJt = Φ(z̄Jt − σJt

)+ ω̄Jt

(1−Φ

(z̄Jt))

, (104)

19 Note that this condition has to hold for each state of nature and at all times. When coding

GIMF it has to be coded for time t rather than time t + 1.

28

GJt = Φ(z̄Jt − σJt

), (105)

Γ′J,t = 1−Φ(z̄Jt), (106)

G′J,t = ω̄Jt f(ω̄Jt). (107)

As for the evolution of entrepreneurial net worth, we first note that banks make zero profits at all

times. The difference between the aggregate returns to capital net of bankruptcy costs and the sum of

deposit interest paid by banks to households therefore goes entirely to entrepreneurs and accumulates.

To rule out a situation where over time so much net worth accumulates that entrepreneurs no longer

need any loans, we assume that they regularly pay out to households dividends which, in terms

of sector J output, are given by divJt . Net worth is also subject to output destroying shocks

SJ,nwyshkt . We assume that for an individual entrepreneur both dividends and output destroying

shocks are proportional to his net worth, which given our above result concerning the proportionality

of borrowing and capital purchases to net worth implies that the evolution of aggregate net worth is

a straightforward aggregation of the evolution of entrepreneur specific net worth. Nominal aggregate

net worth therefore evolves as

NJt = retJk,tQ

Jt−1Ǩ

J

t−1

(1− µJt GJt

)− ǐt−1BJt−1 − P J̃t

(divJt + S

J,nwyshkt

). (108)

This can be combined with the aggregate version of the balance sheet constraint (83) and normalized

to yield

ňJt =řtgn

ňJt−1 + qJt−1Ǩ

J

t−1

(rětJk,tgn

(1− µJt GJt

)− řtgn

)

− pJ̃t(ďivJt + Š

J,nwyshkt

). (109)

Dividends in turn are given by the following expressions:

ďivJt = iňcJt + θ

Jnw

(ňJt − ňJ,filtt

)(110)

pJ̃t iňcJt = Et

SJ,nwdt(kincJh − kincJl + 1

)ΣkincJh

k=kincJl

[ňJt+j + p

J̃t+j

(dǐvJt+j + Š

J,nwyshkt+j

)], (111)

ňJ,filtt = EtΣknwhk=knwl

(ňJt+j

)/ (knwh − knwl + 1) . (112)

Regular dividends, given by expression (111), are a fraction SJ,nwdt (with S̄J,nwd typically in a range

between 0 and 0.05) of smoothed (moving average) gross returns on net worth invested in the previous

29

period, as per equation (109), with kincJh /kincJl the maximum lead/lag of the moving average. The

dividend related net worth shock SJ,nwdt can cause temporary losses or gains of net worth that are

a pure redistribution between households and entrepreneurs, without direct resource implications.

The second determinant of dividends in (110) consists of a dividend response to deviations of net

worth from its long-run value, the latter proxied by a moving average of past and future values of

net worth. This allows us to model dividend policy as a tool to rebuild net worth more quickly

following a negative shock. The parameter θJnw (typically in a range between 0 and 0.05) measures

the increase/decrease in dividends if net worth rises/falls below its long-run value. The relative price

pJt enters because dividends are in units of sector J output while net worth is in units of final output.

We define

ďEPt = pNt ďiv

Nt + p

THt ďiv

Tt . (113)

Output and capital destroying net worth shocks are easier to calibrate if they are expressed as

fractions of steady state net worth.20 We therefore adopt the definitions

ŠJ,nwyt =pJ̃t Š

J,nwyshkt

n̄J, (114)

ŠJ,nwkt =

ŠJ,nwkshkt qJt

((ǨJ

t−1

)/ (gn)

)

n̄J, (115)

and express the shock processes as autocorrelated shocks to ŠJ,nwyt and ŠJ,nwkt .

Finally, we define the sector J bankruptcy cost, which has to be paid out of the output of sector

J , as

rbrJt =

ǨJ

t−1

gn

(rětJk,tq

Jt−1µ

Jt GJt

)

pJt. (116)

8 Raw Materials Producers

In each period each country receives an endowment flow of raw materials X̌supt that, in the

absence of exogenous shocks, is constant in normalized terms (i.e. it grows at the rate g). This

endowment is sold to manufacturers worldwide, with total demand for each country given by X̌demt .

20 Dividend related shocks are easier to calibrate as they are already in terms of a share of

gross returns on net worth.

30

The value of a country’s normalized raw materials exports is therefore given by

X̌xt = pXt (X̌

supt − X̌demt ). (117)

The world market for raw materials is perfectly competitive, with flexible prices that are arbitraged

worldwide. A constant share sxd of steady state (after normalization) raw materials revenue is paid

out to domestic factors of production as dividends d̄X . The rest is divided in fixed shares (1 − sxf )and sxf = Σ

Ñj=2s

xf (1, j) between payments to the government ǧ

Xt , for the case of publicly owned

producers, and dividends to foreign owners in all other countries f̌Xt . This means that all benefits

of favorable raw materials price shocks accrue exclusively to the government and foreigners, and

vice versa for unfavorable shocks. This corresponds more closely to the situation of many countries’

raw materials sectors than the polar opposite assumption of assuming equal shares between the three

recipients at all times. We have

d̄X = sxd p̄XX̄sup , (118)

f̌Xt (1, j) = sxf (1, j)

(pXt X̌

supt − d̄X

), (119)

f̌Xt = f̌Xt (1) = Σ

Ñj=2f̌

Xt (1, j) , (120)

ǧXt = pXt X̌

supt − d̄X − f̌Xt , (121)

where by international arbitrage we have

pXt = pXt (Ñ)et . (122)

The dividends received by country 1 households from ownership of country j raw materials producers

are then given by

ďFt (1, j) = f̌Xt (j, 1)

et(1)

et(j), (123)

and aggregate dividends are

ďFt = ďFt (1) = Σ

Ñj=2ď

Ft (1, j) . (124)

The raw materials sector is subject to shocks to domestic supply X̌supt and to foreign demand, the

latter via the raw materials share parameter in the manufacturing (αXJt) and retail (αXCt

) sectors. Total

31

demand for each country is given by

X̌demt = X̌Tt + X̌

Nt + X̌

Ct , (125)

where X̌Ct is demand from the retail sector, that is directly from household consumption. The market

clearing condition for the raw materials sector is worldwide, and given by

ΣÑj=1

(X̌sup(j)t − X̌

dem(j)t

)= 0 . (126)

9 Unions

There is a continuum of unions indexed by i ∈ [0, 1]. Unions buy labor from households and selllabor to manufacturers. They are perfectly competitive in their input market and monopolistically

competitive in their output market. Their wage setting is subject to nominal rigidities. We first

analyze the demands for union output and then describe their optimization problem.

Demand for unions’ labor output varieties comes from manufacturing firms z ∈ [0, 1] in sectorsJ ∈ {N,T}. The demand for union labor by firm z in sector J is given by a CES production functionwith time-varying elasticity of substitution σUt ,

UJt (z) =

(∫ 1

0

(UJt (z, i)

)σUt−1σUt di

) σUtσUt

−1

, (127)

where UJt (z, i) is the demand by firm z for the labor variety supplied by union i. Given imperfect

substitutability between the labor supplied by different unions, they have market power vis-à-vis

manufacturing firms. Their demand functions are given by

UJt (z, i) =

(Vt(i)

Vt

)−σUtUJt (z) , (128)

where Vt(i) is the wage charged to employers by union i and Vt is the aggregate wage paid by

employers, given by

Vt =

(∫ 1

0Vt(i)

1−σUtdi

) 11−σUt

. (129)

The demand (128) can be aggregated over firms z and sectors J to obtain

Ut(i) =

(Vt(i)

Vt

)−σUtUt , (130)

32

where Ut is aggregate labor demand by all manufacturing firms.

GIMF allows for three types of wage rigidities. The first two are the conventional cases of nominal

wage rigidities. Sticky wage inflation takes the form familiar from (42):

GUP,t(i) =φPU

2UtTt

Vt(i)Vt−1(i)

Vt−1Vt−2

− 1

2

. (131)

Note that these adjustment costs are zero in steady state even though real wages grow at the rate

of world technological progress. Also, the level of world technology enters as a scaling factor in

(131), as otherwise these costs would become insignificant over time. The second type of wage

rigidities is real wage rigidities, whereby unions resist rapid changes in the real wage Vt/Pct . We

define πrwt (i) = πvt (i)/

(gπCt

). Then these adjustment costs are given by

GUP,t(i) =φPU

2UtTt (π

rwt (i)− 1)2 =

φPU

2UtTt

Vt(i)Vt−1(i)

gπCt− 1

2

. (132)

The stochastic wage markup of union wages over household wages is given by µUt = σUt/(σUt −1).The optimization problem of a union consists of maximizing the expected present discounted

value of nominal wages paid by firms Vt(i)Ut(i) minus nominal wages paid out to workers WtUt(i),

minus nominal wage inflation adjustment costs PtGUP,t(i). Unlike manufacturers, this sector does not

face fixed costs of operation. It is assumed that each union pays out each period’s nominal net cash

flow as dividends DUt (i). The objective function of unions is

Max{Vt+s(i)}

∞

s=0

EtΣ∞s=0R̃t,s

[(Vt+s(i)−Wt+s)Ut+s(i)− Vt+sGUP,t+s(i)

], (133)

subject to labor demands (130) and adjustment costs (131) or (132). We obtain the first order

condition for this problem. As all unions face an identical problem, their solutions are identical

and the index i can be dropped in all first-order conditions of the problem, with Vt(i) = Vt and

Ut(i) = Ut. We let πVt = Vt/Vt−1, the gross rate of wage inflation, and we rescale by technology.

For nominal wage rigidities we obtain the condition[µUt

w̌tv̌t− 1]

= φPU(µUt − 1

)(

πVtπVt−1

)(πVtπVt−1

− 1)

(134)

−Etθgn

řt+1φPU

(µUt − 1

) v̌t+1v̌t

Ǔt+1

Ǔt

(πVt+1πVt

)(πVt+1πVt

− 1)

.

33

For real wage rigidities we have[µUt

w̌tv̌t− 1]

= φPU(µUt − 1

)πrwt (π

rwt − 1) (135)

−Etθgn

řt+1φPU

(µUt − 1

) v̌t+1v̌t

Ǔt+1

Ǔtπrwt+1

(πrwt+1 − 1

).

Real “dividends” from union organization, denominated in terms of final output, are distributed

lump-sum to households in proportion to their share in aggregate labor supply. After rescaling they

take the form

ďUt = (v̌t − w̌t)Ǔt − v̌tǦUP,t . (136)We also have v̌t/v̌t−1 = (Vt/PtTt)/(Vt−1/Pt−1Tt−1), so that

v̌tv̌t−1

=πVtπtg

. (137)

Finally, the labor market clearing condition equates the combined labor supply of OLG and LIQ

households to the labor demands coming from nontradables and tradables manufacturers, including

their respective labor adjustment costs if applicable, and from unions for wage adjustment costs. We

have:

Ľt = ǓNt + Ǔ

Tt + Ǧ

NU,t + Ǧ

TU,t + Ǧ

UP,t . (138)

10 Import Agents

Each country, in each of its export destination markets, owns two continua of import agents, one

for manufactured intermediate tradable goods (T ) and another for final goods (D), each indexed

by i ∈ [0, 1] and by J ∈ {T,D}. Import agents buy intermediate goods (or final goods)from manufacturers (or distributors) in their owners’ country and sell these goods to distributors

(intermediate goods) or consumption/investment goods producers (final goods) in the destination

country. They are perfectly competitive in their input market and monopolistically competitive in

their output market. Their price setting is subject to nominal rigidities. We first analyze the demands

for their output and then describe their optimization problem.

34

Demand for the output varieties supplied by import agents comes from distributors (sector T ) or

consumption/investment goods producers (sectors D), in each case indexed by z ∈ [0, 1]. Recall thatthe domestic economy is indexed by 1 and foreign economies by j = 2, ..., Ñ . Domestic distributors

z require a separate CES imports aggregate Y JMt (1, j, z) from the import agents of each country

j. That aggregate consists of varieties supplied by different import agents i, Y JMt (1, j, z, i), with

respective prices P JMt (1, j, i), and is given by

Y JMt (1, j, z) =

(∫ 1

0

(Y JMt (1, j, z, i)

)σJM−1σJM di

) σJMσJM−1

. (139)

This gives rise to demands for varieties of

Y JMt (1, j, z, i) =

(P JMt (1, j, i)

P JMt (1, j)

)−σJMY JMt (1, j, z) , (140)

P JMt (1, j) =

(∫ 1

0P JMt (1, j, i)

1−σJMdi

) 11−σJM

, (141)

and these demands can be aggregated over z to yield

Y JMt (1, j, i) =

(P JMt (1, j, i)

P JMt (1, j)

)−σJMY JMt (1, j) . (142)

Nominal rigidities in this sector take the form familiar from (42),

GJMP,t (1, j, i) =φPJM

2Y JMt (1, j)

P JMt (1,j,i)P JMt−1 (1,j,i)

P JMt−1 (1,j)

P JMt−2 (1,j)

− 1

2

, (143)

and represent a claim on the underlying exports. Import agents’ cost minimizing solution for

inputs of manufactured intermediate tradable goods (or final goods) varieties therefore follows

equations (36) - (38) above (or similar conditions for demands of consumption/investment goods

producers). We denote the price of inputs imported from country j at the border of country 1 by

P JM,cift (1, j), the cif (cost, insurance, freight) import price. By purchasing power parity this satisfies

P JM,cift (1, j) = p̃expt P

JHt (j)Et(1)/Et(j), where p̃expt is an exogenous price shock that equals the

inverse of a shock to the technology that converts foreign exports into domestic imports. In real

terms we have

pJM,cift (1, j) = pJHt (j)p̃

expt (j)

et(1)

et(j). (144)

35

The optimization problem of import agents consists of maximizing the expected present

discounted value of nominal revenue P JMt (1, j, i)YJMt (1, j, i) minus nominal costs of inputs

P JM,cift (1, j)YJMt (1, j, i), minus nominal inflation adjustment costs PtG

JMP,t (1, j, i). The latter

represent a demand for final output. This sector does not face fixed costs of operation. It is assumed

that each import agent pays out each period’s nominal net cash flow as dividends DJMt (1, j, i). The

objective function of import agents is

Max{PJMt+s (1,j,i)}∞s=0

EtΣ∞s=0R̃t,s

[(P JMt+s (1, j, i)− P JM,cift+s (1, j)

)Y JMt+s (1, j, i)− P JMt+s GJMP,t+s(1, j, i)

],

(145)

subject to demands (142) and adjustment costs (143). The first order condition for this problem, after

dropping firm specific subscripts and rescaling by technology, has the form:

[σJM

σJM − 1pJM,cift (1, j)

pJMt (1, j)− 1]

=φPJM

σJM − 1

(πJMt (1, j)

πJMt−1(1, j)

)(πJMt (1, j)

πJMt−1(1, j)− 1)

(146)

−Etθgn

řt+1

φP JM

σJM − 1pJMt+1(1, j)

pJMt (1, j)

Y̌ JMt+1 (1, j)

Y̌ JMt (1, j)

(πJMt+1(1, j)

πJMt (1, j)

)(πJMt+1(1, j)

πJMt (1, j)− 1)

.

The rescaled real dividends of country j’s import agent in the domestic economy, which are paid out

to OLG households in country j, are

ďJMt (1, j) = (pJMt (1, j)− pJM,cift (1, j))Y̌ JMt (1, j)− pJMt (1, j)ǦJMP,t (1, j) . (147)

The total dividends received by OLG households in country 1, expressed in terms of country 1

output, are

ďJMt = ďJMt (1) = Σ

Ñj=2ď

JMt (j, 1)

et(1)

et(j), (148)

ďMt = ďTMt + ď

DMt . (149)

Finally, the market clearing conditions for import agents equate the export volume received from

abroad to the import volume used domestically plus adjustment costs:

Y̌ JXt (j, 1) = Y̌JMt (1, j) + Ǧ

JMP,t (1, j) . (150)

36

11 Distributors

Distributors produce domestic final output. They buy domestic tradables and nontradables from

domestic manufacturers, and foreign tradables from import agents. They also use the stock of public

infrastructure free of a user charge. Distributors sell their final output composite to consumption

goods producers, investment goods producers and final goods import agents in foreign countries.

They are perfectly competitive in both their output and input markets.

We divide our description of the technology of distributors into a number of stages. In the first

stage a foreign input composite is produced from intermediate manufactured inputs originating in all

foreign economies and sold to distributors by import agents. In the second stage a tradables composite

is produced by combining these foreign tradables with domestic tradables, subject to an adjustment

cost that makes rapid changes in the share of foreign tradables costly. In the third stage a tradables-

nontradables composite is produced. In the fourth stage the tradables-nontradables composite is

combined with a publicly provided stock of infrastructure.

Foreign input composites Y JFt (1), J ∈ {T,D}, are produced by combining imports Y JMt (1, j)originating in different foreign economies j and purchased through import agents. A foreign input

choice problem therefore only arises when there are more than 2 countries. Also, distributors use

only the composite indexed by T , while the composite indexed by D is used by consumption and

investment goods manufacturers. We present the problem here in its general form and then reapply

the results when describing these other agents. The CES production function for Y JFt (1) has an

elasticity of substitution ξJM and share parameters ζJ(1, j) that are identical across firms and that

add up to one, ΣÑj=2ζJ(1, j) = 1. We also allow for an additional effect of technology shocks on

the intermediates import share parameters. Specifically, we posit that an improvement in technology

in a foreign country not only leads to a lower cost in that country, but also to a higher demand for

the respective good in all foreign countries, reflecting quality improvements due to better technology.

The import share parameter between countries 1 and j is therefore given by

ζ̃T

(1, j) =

(ζT (1, j)ATt (j)

κ(1)

ζ̃T

(1)

)

, (151)

ζ̃T

(1) = ΣÑj=2ζT (1, j)ATt (j)

κ(1) , (152)

37

where κ = 0 corresponds to the standard case while κ > 0 introduces positive foreign demand

effects of technological progress. This makes it more likely that technological progress in the

tradables sector will lead to a real appreciation. By contrast, for investment and consumption goods

producers we assume ζ̃D

(1, j) = ζD(1, j). The local currency prices P JMt (1, j) of imports in

country 1 are determined by import agents, and the overall cost of the bundle Y JFt (1) is PJFt (1).

In the calibration of the model the share parameters ζJ(1, j) will be parameterized using a multi-

region trade matrix. We have the following sub-production function:

Y JFt (1) =

(ΣNj=2ζ̃

J(1, j)

1

ξJM

(Y JMt (1, j)

) ξJM−1ξJM

) ξJMξJM−1

, (153)

with demands

Y JMt (1, j) = ζ̃J(1, j)Y JFt (1)

(P JMt (1, j)

P JFt (1)

)−ξJM(154)

and an import price index, written in terms of relative prices, of

pJFt (1) =(

ΣNj=2ζ̃J(1, j)

(pJMt (1, j)

)1−ξJM) 1

1−ξJM . (155)

Equations (153) and (154) are rescaled by technology and population to generate aggregate

foreign input demand of country 1, Y̌ JFt (1) and aggregate demands for individual country imports

Y̌ JMt (1, j). Note that for final goods Y̌DFt there is a market clearing condition because the imported

bundle is sold to both consumption and investment goods producers:

Y̌ DFt = Y̌CFt + Y̌

IFt . (156)

In the two country case equations (153)-(155) simplify, after aggregation, to Y̌ JFt (1) = Y̌JMt (1, 2)

and pJFt = pJMt . In our notation we will now revert to the two-country case and drop the index 1 for

Home.

The tradables composite Y Tt is produced by combining foreign produced tradables YTFt with

domestically produced tradables Y THt , in a CES technology with elasticity of substitution ξT . A key

concern in open economy DSGE models is the potential for an excessive short-term responsiveness of

international trade to real exchange rate movements. This model avoids that problem by introducing

adjustment costs GTF,t that make it costly to vary the share of Foreign produced tradables in total

tradables production Y TFt /YTt relative to the value of that share in the aggregate distribution sector

38

in the previous period Y TFt−1/YTt−1. At the previous level we allowed for the possibility κ > 0,

meaning foreign technology shocks affect relative demands for goods from different countries. We

allow for an identical effect, dependent on the same parameter, to affect relative demands for domestic

and foreign tradable goods. Specifically, an improvement in average world technology increases the

relative demand for foreign produced tradables. The domestic and foreign tradables share parameters

are therefore given by

α̃THt =αTHt

(ATt)κ

ᾰTHt, (157)

α̃TFt =

(1− αTHt

) (ARWt

)κ

ᾰTHt, (158)

ᾰTH = αTHt

(ATt)κ

+(1− αTHt

) (ARWt

)κ, (159)

ARWt = ΣÑj=2A

Tt (j)

gdpss(j)

ΣÑk=2

gdpss(k) . (160)

The sub-production function for tradables then has the following form:21,22

Y Tt =

((α̃THt

) 1ξT

(Y THt

) ξT−1ξT+(α̃TFt) 1ξT

(Y TFt (1−GTF,t)

) ξT−1ξT) ξT

ξT−1

, (161)

GTF,t =φFT

2

(RTt − 1

)2

1 +(RTt − 1

)2 , (162)

RTt =Y TFtY TtY TFt−1Y Tt−1

. (163)

After expressing prices in terms of the numeraire, and after rescaling by technology and population,

we obtain the aggregate tradables sub-production function from (161) - (163). We also obtain the

following first-order conditions for optimal input choice:

Y̌ THt = α̃THt Y̌

Tt

(pTHtpTt

)−ξT, (164)

21 Home bias in tradables use depends on the parameter αTH and on a similar parameterαDH at the level of final goods imports.22 For the ratioRTt we assume as usual that the distributor takes the lagged denominator term as given in hisoptimization.

39

Y̌ TFt[1−GTF,t

]= α̃TFtY̌

Tt

(pTFtpTt

)−ξT (ÕTt

)ξT, (165)

ÕTt = 1−GTF,t − φFTRTt(RTt − 1

)

[1 +

(RTt − 1

)2]2. (166)

The tradables-nontradables composite Y At is produced with another CES production function

with elasticity of substitution ξA. We again allow for a relative demand effect, this time of

nontradables productivity shocks, with input share parameters given by

α̃Tt =(1− αN)ᾰNt

, (167)

α̃Nt =αN(ANt)κ̃

ᾰNt, (168)

ᾰNt = αN(ANt)κ̃

+ (1− αN) . (169)The sub-production function for the tradables-nontradables composite then has the following form:

Y At =

(

(α̃Tt)1

ξA

(Y Tt) ξA−1ξA

+ (α̃Nt)1

ξA

(Y Nt) ξA−1ξA

) ξAξA−1

. (170)

The real marginal cost of producing Y At is, with obvious notation for sectorial price levels,

pAt =[α̃Tt(pTt)1−ξA + α̃Nt

(pNt)1−ξA

] 11−ξA . (171)

After expressing prices in terms of the numeraire, and after rescaling by technology, we obtain the

aggregate tradables-nontradables sub-production function from (170), and the following first-order

conditions for optimal input choice:

Y̌ Nt = α̃Nt Y̌At

(pNtpAt

)−ξA, (172)

Y̌ Tt = α̃Tt Y̌At

(pTtpAt

)−ξA. (173)

For the case where the nontradables sector is excluded from GIMF, we simply have Y̌ At = Y̌Tt and

pAt = pTt .

40

The private-public composite ZDt , which we will refer to as domestic final output, is produced

with the following production function:

ZDt = YAt

(KG1t

)αG1 (KG2t

)αG2 S . (174)

The inputs are the tradables-nontradables composite Y At and the stocks of public capital KG1t and

KG2t , which are identical for all firms and provided free of charge to the end user (but not of course

to the taxpayer). Note that this production function exhibits constant returns to scale in private inputs

while the public capital stocks enter externally, in an analogous manner to exogenous technology.

The term S is a technology scale factor that can be used to normalize steady state technology to one,(K̄G1

)αG1 (K̄G2)αG2 S = 1.

The real marginal cost of ZDt is denoted as pDHt , while the real marginal cost of Y

At is p

At . After

expressing prices in terms of the numeraire, and after rescaling by technology and population, we

obtain the normalized production function from (174), and the follow